Understanding Derivatives: dy/dx vs. d/dx

This video tutorial explains the difference between dydx and d/dx in the context of derivatives. It covers how dydx represents the derivative of a function, while d/dx is a command to differentiate. The video demonstrates using the power rule to find derivatives and introduces implicit differentiation. It also discusses related rates and differentiating with respect to time. A pop quiz is included to test understanding.

What is the primary difference between dy/dx and d/dx?

• Both are statements.

• dy/dx is a command, d/dx is a statement.

• dy/dx is a statement, d/dx is a command.

• Both are commands.

When you see d/dx, what should you do?

• Differentiate the function with respect to x.

• Integrate the function.

• Divide the function by x.

• Multiply the function by x.

Using the power rule, what is the derivative of x^3 with respect to x?

• 3x^3

• x^3

• x^2

• 3x^2

What additional factor appears when differentiating y^3 with respect to x?

• 3y^2

• 3x^2

• dy/dx

• 3y^2 * dy/dx

What is implicit differentiation used for?

• Differentiating explicit functions.

• Differentiating constant functions.

• Differentiating functions where variables are not isolated.

• Differentiating functions with respect to time.

How do you differentiate x^5 with respect to a variable s?

• 5x^4

• 5x^4 * dx/ds

• 5x^5 * ds/dx

• x^5 * ds/dx

In related rates problems, what does d/dt represent?

• Multiplication by time.

• Integration with respect to time.

• Differentiation with respect to time.

• Differentiation with respect to x.

What is the derivative of x^4 + y^3 with respect to time?

• 4x^3 * dt/dx + 3y^2 * dt/dy

• 4x^3 * dx/dt + 3y^2 * dy/dt

• x^4 + y^3

• 4x^3 + 3y^2

In the pop quiz, what is the derivative of x^3 with respect to y?

• 3x^2 * dx/dy

• 0

• x^3

• 3x^2

What is the derivative of t^5 with respect to y in the pop quiz?

• 0

• 5t^4 * dt/dy

• 5t^4

• t^5